Modeling, Simulation, and Membrane Wetting Estimation in Gas–Liquid Contacting Processes Including Shell-Side Reaction: Biogas Upgrading Using DEA Solution

The basic principles of a steady-state mass transfer model and the resistance-in-series film model are assessed with the aid of a series of experiments in a gas–liquid contact membrane mini-module (3 M Liqui-Cel MM-1.7 × 5.5) using an aqueous solution of diethanolamine (DEA) of 0.25 M (mol/L) for biogas upgrading. Experimental data show that CO2 removal may exceed 67% and reach 100% in combination with the highest possible recovery of CH4 when employing biogas flow rates in the range of 2.8 × 10–5 – 3.6 × 10–5 m3/s and solvent flow rates within 0.47 × 10–5 – 0.58 × 10–5 m3/s. For the experimental data set, a correlation has been developed, effectively interpolating CO2 removal with the gas and liquid flow rates. The wetting values calculated are concentrated close to each other for the same liquid flow rate without considerably depending on the gas flow rate, especially when applying the Hikita–Yun (reaction rate–shell-side correlation) compared with the Hikita–Costello pair. Furthermore, the calculated wetting diminishes with increasing liquid flow rate, a result that is consistent with previous modeling attempts and relevant literature indications. The assumption of enhanced mass transfer in the liquid-filled part of the membrane pores due to the reaction is scrutinized, leading to objectionable computational wetting values. It is shown that for a concentration of DEA equal to 0.25 M the Hatta numbers and the enhancement factors are not equal in the whole reaction path; thus, the choice of the shell-side correlation has an appreciable impact on the overall analysis, especially for the determination of the wetting values.


Appendix A -Additional equations for the membrane-based gas absorption system
For the sake of comparison as part of the membrane modelling canon the following dimensionless quantities are introduced in the current study analysis: ̂= /  , ̂=  •   •  • /(4 , ) = 1  ⁄ , ℎ  = 2    /,  * =  , /  (S.1) , () is the lumen mixed-cup concentration for every i component defined as: , () = ∫ 2 •   (, ) (S. 2) It has to be noted that the ̂ and the  * parameters are reference values applying the feed (constant) gas mixture flowrate,  , , unlike the lumen-side gas mixture average velocity, , which is defined taking into account the variation of the gas flowrate in the fibers due to pressure drop and mainly the loss of CO2 where  , ,  , are the gas mixture flowrates as measured by the gas flowmeter at the inlet and outlet of the membrane module, respectively, and  , is the averaged gas flowrate.
The lumen-side gas mixture average velocity is then found to be: where   is the number of fibers.(Note that for ease of calculations the averaged gas flowrate can be taken equal to the mean value of the inlet and outlet flowrates without much difference).
Regarding flow through a bundle of capillary tubes the hydraulic diameter ( ℎ ) concept stems from the Carman-Kozeny theory and it is related to the permeability, : where   is the Kozeny constant,  is the void fraction of the bed (module) so that it holds  = 1 −  (see the main article, Table 1 for the packing fraction, ), and  0 is a parameter referring to the geometry of the tubes or particles, with units 1/m [S1].
Following Eq. (S.5) the hydraulic diameter is given by: In Eq. (S.6) the volumetric area adjusted to the solid structure, A0, is introduced such that A0=4/dcirc for the case of circular cylinder particles of diameter dcirc, and A0=6/dp for the case of spherical particles of diameter dp [S1].The interstitial (i.e.within the voids) velocity,   , is related to the superficial (i.e. for a chamber without tubes or particles) velocity,   , as: For example, given the shell-side volumetric flowrate (i.e. the one measured by the rotameter) the superficial velocity may be given by dividing with the whole cross-section of the module (i.e. the one given by the manufacturerssee the main article, Table 1).What is needed is the interstitial velocity in the expressions for the dimensionless numbers, e.g.Reynolds number [S2] where   is the diameter of the module,  , ,  , are the outer and inner diameters of one fiber, respectively,   is the voids volume (shell-side volume accounting for the presence of the bunch of fibers),   the inner volume of the module as if it were without fibers, and   (units: 1/m) is the ratio of the effective mass transfer area (based on the inner fiber diameter),    , to the shell-side voids volume,   .
The 1-D postulation in the shell side may resemble the mass transfer behavior in the parallel-flow module used in this study in the absence of significant radial gradients.Computationally, the shell-side mass transfer behavior can be obtained by an 1-D model if all source terms transverse to the parallel flow (i.e. from the fiber zone to the shell side) are included (see e.g.Koutsonikolas et al. [S3] for a similar treatment).
For a 2-D shell-side formulation analytical solutions for the velocity profile for parallel-flow modules using Happel's model and Dirichlet BC at the outer diameter of the fiber can be found in [S4].Happel's free surface model [S5] presupposes that the arrangement of the bundle of cylinders (here: fibers) in a volume (e.g.triangular, square arrays) and the radius of the free-surface postulation are knownsee also [S6, S7, S8, S9, S10, S11].Otherwise, the 1-D model contains all the relevant transfer phenomena that would have been included in a model of higher geometric complexity without further assumptions regarding the shellside configuration.
The removal efficiency of the absorption process is given by the difference of the gas flowratesy referring to CO2:

Appendix B -Physico-chemical properties
Viscosities and binary diffusivities in the lumen side,  , , are calculated according to the methodology presented in [S12].In order to calculate the effective diffusion coefficients in the membrane mesoporous zone of the gas-liquid contact membrane processin the gas separation process only macroscopic permeance values are neededit is crucial to calculate Knudsen diffusion coefficients following the resistance-in-series formula [S13]: where  is the tortuosity and   is the porosity of the porous network (provided by the membrane modules supplier).
The Henry's constants for the system CO2aqueous solutions of amines which are inserted to Eq. ( 6) of the main article are calculated using the N2O-CO2 analogy [S14, S15, S16]: where   2 , is the Henry's constant of N2O in water and   2 , is the Henry's constant of CO2 in water,   2 , is the Henry's constant of N2O in the aqueous amine solution and   2 , is the Henry's constant of CO2 in the aqueous amine solution.
The Henry's constant of N2O and CO2 in water as a function of temperature is given by [S16] see Table S.1: , = (  +   / +    +   ) for i=N2O, CO2,  , in Pa m 3 /mol, T in K (S.14) The Henry's constant of N2O in pure diethanolamine (DEA) as a function of temperature is given by [S16]: The semi-empirical expression for the Henry's constant of N2O in aqueous DEA solutions is given bysee  In this study in the absence of relevant literature data the Henry's constant for the system CH4-DEA is taken as that of the corresponding constant of CH4 into water, which in any case overestimates the solubility of methane when compared to the presence of an electrolyte or an alkaline solution [S17].At 25 0 C the Henry's constant for the system CH4-DEA (~CH4-H2O) is taken equal to 0.658 Atm .m 3 /mol [S18], a value almost similar to those presented by Sander in the author's literature review [S19].
Generally, the temperature dependence of the diffusion coefficients in water can be well approximated by: where   is the activation energy for diffusion in water.Table S.2 summarizes the preexponential and the activation energy values for the diffusion coefficients of the two gases in water.The diffusivity of CH4 in aqueous solutions of DEA can be assumed to be equal to the diffusivity in water, while the diffusivity of CO2 in aqueous solutions of DEA is derived by [S22]:

Appendix C -Additional literature review, figures and comments
The gas-liquid contact membrane process considers a mixture (gas or liquid) flowing in the lumen (fiber side), which is confined by the membrane not allowing direct contact of the mixture (liquid or gas) flowing in the outer part of the membrane, in the shell side.The degree of the separation of the mixture into its constituents for this kind of process depends on whether there are reactive conditions between the gas species and the solvent, and not on the membrane which does not exert any particular selectivity to one species over the other [S24, S25].
In the past, for the absorption of CO2 and especially biogas upgrading various membrane materials have been used in gas-liquid contact membrane processes, such as polypropylene (PP) [S7, S26, S27, S28, S30, S31, S32, S33, S34, S35, S36], polytetrafluoroethylene (PTFE) [S9, S25, S37, S38], polyvinylidenefluoride (PVDF) [S7, S39], polydimethylsiloxane [S40], or mixed-matrix membranes [S41], using H2O [S25, S26, S32, S40see also reviews [S25, S42] and references mentioned therein], NaOH/KOH [S26, S33], NH3 [S38], ionic liquids [S39] and others [S9, S31, S34, S35, S36, S37] (see also a review by Pantoleontos et al. [S29] of membrane materials and solvents for CO2 capture).Amine solvents such as the primary amine monoethanolamine (MEA), the secondary amine DEA and the tertiary amine methyldiethanolamine (MDEA), which are preferred in conventional packed towers for CO2 capture [S43] (see also the evaluation of over 130 aqueous amine solvents regarding carbon-capture performance by Bernhardsen & Knuutila [S30]), are extensively reviewed in membrane-based absorption processes; see e.g.MEA or triethanolamine (TEA) using PVDF membranes [S44], MEA using PP [S45, S46], MEA or DEA using PTFE membranes [S47], blended dimethylethanolamine (DMEA)/MEA in PTFE membranes [S48], aqueous solutions of DEA [S27, S30, S7, S41], MDEA, piperazine (PZ) and 2-amino-2-methyl-1-propanol (AMP) [S31], activated MDEA in PP membranes [S49], investigation of tertiary amines (2-(diethylamino)ethanol - The enhancement factors in the overall analysis serve as the effect of the chemical reaction on the absorption rate and in principle can be derived by solving a set of equations in the concentration boundary layer (here: of the shell side) where the gaseous components are absorbed in a combination of mechanisms of diffusion and reaction before transported to the liquid bulk [S69].Thus, the calculation of the dissolved gas concentration involves the solution of the coupled problem of the concentration boundary layer model with the macroscopic model for the liquid bulk including consistent boundary conditions at the computational interfaces [S70].In the corresponding analysis, the role of different mass transfer models (e.g.film (Nernst-Whitman); penetration and surface-renewal (Higbie and Danckwerts) models [S57]) may not be so crucial since the maximum discrepancy noticed is only a few percent [S57] still, the determination of the computational wetting depends on the definition of the membrane mass transfer model, see Eqs. ( 9) and ( 10) of the main manuscript.Analytical expressions for enhancement factors for second-order reversible reactions [S71] or approximations for n th -order reversible reactions [S72] are available in the literaturesee also discussion in [S69, S73, S74, S75].
Figure S1 illustrates the piping and instrumentation diagram of the experimental gas-liquid contact membrane process as adapted from [S76].The unit setup can be operated either with liquid recycle representing a semi-batch operation mode or on a once-through mode representing a continuous operation mode.The liquid solvents preparation takes place in a 6 L Stainless Steel (SS316) feed tank equipped with a pressure gauge and safety valve.Liquid solvent is being fed with a high precision gear pump (Ismatec ISM446B) through a float ball flowmeter (0-0.5 L/min).Through a 3-way valve, the liquid phase either recirculates into the mixing vessel or is directed to the membrane contactor section.Feed gas is being supplied through two different compressed gas cylinders (containing either single gases or gas mixtures) using two independent Mass Flow Controllers (MFCs) (Bronkhorst F-201CV-20K-AAD-22-V, 1L/min, 5 bar (g)/3 bar (g), CO2 and Bronkhorst F-201CV-20K-AAD-22-V, 1L/min, 5 bar (g)/3 bar (g), N2) at flow rates up to approximately 2 L/min (std).Through a series of valves, the feed gas can be either sent directly to the residue/analysis equipment for feed flow and composition measurements or to the membrane module and afterwards to the analysis equipment.
Figure S2 depicts the total membrane mass transfer resistance for CO2 as calculated by Eq. ( 8) of the main manuscript with wetting values estimation when applying the Hikita-Costello or the Hikita-Yun pair.A slight deviation from a straight line is due to the slightly different temperatures of the performed experiments.It has to be noted that the individual resistances, gas and liquid, do not depend on the extent of the calculated wetting (thus, on the shell-side mass transfer correlation used) since they are defined as if the whole length of the pore had been filled with either gas or liquid, respectivelysubsequently, the estimated wetting value from each correlation is applied to Eq. ( 8) of the main manuscript to derive the effective membrane mass transfer coefficient,  , , whose inverse value,   , is the total membrane mass transfer resistance (for CO2 in Table 4 of the main manuscript).

Figure S2 .
Figure S2.Linear relationship of the total membrane mass transfer resistance of CO2,   , with the wetting for the Hikita-Costello (a) and the Hikita-Yun (b) pairs.

Table S .
1: Parameters for the Henry's constant of N2O in water and pure amines, for CO2 in water, and the binary mixture[S16].

Table S
.2: Pre-exponential and activation energy values for calculation of gaseous diffusivities in water.